Define and use set notation of number systems, including integers (ℤ), positive integers (ℤ+), negative integers (ℤ−), rational numbers (ℚ), irrational numbers (ℚ′), and real numbers (ℝ).

Consider the function $f(x) = \sqrt{x^2 - 9}$ defined on the set $A = \{x \in \mathbb{R} : x^2 - 9 \geq 0\}$. (a) Express the set $A$ in interval notation. (1 mark) (b) Classify the number $f(5)$ by identifying which of the following number systems it belongs to: $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{Q}'$. Justify your answer. (1 mark) (c) Determine whether $f(A) \subseteq \mathbb{Q}$. Justify your answer by providing a counterexample if necessary. (1 mark)

Consider the Venn diagram showing the relationships between number sets. Determine which statement correctly describes the shaded region.

Consider the set $S = \{x : x^2 = 5\}$. (a) Determine all elements of $S$ and express them in exact form. (b) State which standard number system(s) contains all elements of $S$. (c) Show that no element of $S$ belongs to $\mathbb{Q}$ by using a proof by contradiction for the positive element.

Consider the set $S = \{x : x^2 - 5 = 0, x \in \mathbb{R}\}$. Determine which number system classification correctly describes all elements of $S$.